Question

classify (√7)·(3) as rational or irrational, and explain your reasoning. a suppose an irrational times a nonzero rational = rational, then irrational = rational/rational. this means irrational = rational. thus the irrational number √7 times the rational number 3 equals 3√7, which is both rational and irrational. b suppose an irrational times a nonzero rational = rational, then irrational = rational/rational. this means irrational = rational, which is a contradiction. thus the irrational number √7 times the rational number 3 equals the rational number 3√7. c suppose an irrational times a nonzero rational = rational, then irrational = rational/rational. this means irrational = rational, which is a contradiction. thus the irrational number √7 times the rational number 3 equals the irrational number 3√7. d suppose an irrational times a nonzero rational = rational, then irrational = rational/rational. this means irrational = rational, which is a contradiction. thus the irrational number √7 times the rational number 3 equals 3√7, which is neither rational nor irrational.

Explanation:

• First, assume that an irrational number times a non - zero rational number is rational. Let the irrational number be \(x\) and the non - zero rational number be \(y=\frac{a}{b}\) (\(a,b\in\mathbb{Z},b

eq0,a
eq0\)), and if \(xy = z\) (where \(z\) is rational, \(z = \frac{c}{d}\), \(c,d\in\mathbb{Z},d
eq0\)). Then \(x=\frac{z}{y}=\frac{bc}{ad}\), which would imply that \(x\) is rational, a contradiction.

• \(\sqrt{7}\) is an irrational number (because it cannot be written as a fraction of two integers). 3 is a non - zero rational number (\(3=\frac{3}{1}\)).
• By the property that an irrational number times a non - zero rational number is irrational, \(\sqrt{7}\times3 = 3\sqrt{7}\) is irrational.

Answer:

C. Suppose an irrational times a nonzero rational = rational, then irrational=\(\frac{\text{rational}}{\text{rational}}\). This means irrational = rational, which is a contradiction. Thus the irrational number \(\sqrt{7}\) times the rational number 3 equals the irrational number \(3\sqrt{7}\).